Obama also said that the health care plan will work in the long run, that problems with the website will be fixed, and that people who have had their insurance policies canceled will be helped.

"I continue to believe, and (I am) absolutely convinced that at the end of the day, people are going to look back at the work we've done to make sure that in this country, you don't go bankrupt when you get sick, that families have that security," Obama said. "That is going be a legacy I am extraordinarily proud of."

ABC released clips from the Walters interview that airs Friday night on 20/20.

The website HealthCare.gov has had numerous technical glitches since its unveiling on Oct. 1. Obama administration officials have vowed a well-functioning website by the end of November, which is Saturday.

Many consumers have seen their insurance policies bought on the individual market canceled because they do not meet the standards of the new law. The health care disputes have taken a toll on Obama's political standing, with his approval ratings hitting low points in a string of recent polls.

Some surveys indicate that many voters no longer trust Obama. He took issue with that analysis during his interview with Walters, saying that every president goes through rough patches.

"I got re-elected in part because people did think I was trustworthy and they knew I was working on their behalf," Obama said. "Very rarely are the good things that happen (going to) get the same attention as the things that aren't working so well."

With a little more than three years left in his second term, Obama vowed to push on with his agenda.

"Every president in their second term is mindful that you've only got a limited amount of time," he said, "and you want to make sure you are squeezing every last ounce of energy that you have to try to deliver on the commitments you made to the American people."